ECON 3818

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# ECON 3818
## Chapter 2
### Kyle Butts
### 21 July 2021

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### Chapter 2: Describing Distribution with Numbers

---
## Chapter Overview
	
	
- Population vs. Sample
- Measures of Central Tendency
- Mean
- Median
		      
- Measures of Variability
- Quartiles
- Variance \& Standard Deviation

---
# Population vs Sample
	
	
- .hi.purple[Population]: the entire entities under the study
- Examples: all men, all NBA players, all children under 5
		      
- .hi.green[Sample]: subset of the population
- Can be used to draw inferences about the population
- Examples: our class, Denver Nuggets players, daycares in Colorado
		      
- Interested in parameters of the .hi.purple[population] distribution, we can estimate these parameters using data from .hi.green[samples] since finding population parameters is infeasible

---
# Population Distribution

The following graph depicts the underlying population distribution
	
- We are interested in its parameters, but are unable collect data on every single observation

---
# Population Inference

What we do instead is use a sample of the population and use that sample distribution to determine parameters of interest

.center[
<img style="width:80%;" src="data:image/png;base64,#sample_anim.gif"/>
]

---
# Parameters of Interest
	
	
Two primary .hi.purple[population] parameters of interest:
		      
- Measures of central tendency:
	      	      
	- Population .orange[mean], `$\mu$`
	
	- Population .red_pink[median]
	      	      
- Measures of variability:
	      	      
	- Population .blue[variance], `$\sigma^2$`
	  
We will .it.green[estimate] these using the .hi.green[sample] distribution

---
# Measuring Center: the Mean
	
	
The most common measure of center is the arithmetic average, or .hi.orange[mean]

`$${\color{orange} \bar{x}} = \frac{x_1 + x_2 + .... + x_n}{n}$$`

or more compactly:

`$${\color{orange} \bar{x}}=\frac{1}{n}\sum_{i=1}^n x_i$$`
	
	
---
# Population Inference: Mean

.center[
<img style="width:80%;" src="data:image/png;base64,#sample_anim_mean.gif"/>
]

---
# Measuring Center: the Median
	
	
The .hi.red_pink[median] is the midpoint of a distribution
		      
- Is more resistant to the influence of .hi[extreme observations]
		      
How to calculate median:
		      
- Arrange observations from smallest to largest
- If there is odd number of observations, the median is the center observation. If there are even number of observations, the median is the average of two center observations

---
# Mean vs. Median
	
	
- Although we will primarily be using the mean throughout the semester, the biggest drawback of the mean is that it is not resistant to .hi.purple[outliers]
		      
- The median, however, is resistant to .hi.purple[outliers] so it can be important to calculate for smaller samples

.center[
<img style="width: 60%;" src="data:image/png;base64,#meme.png"/>
]

---
# Mean vs. Median Example

.hi[Median]: 205.5 rebounds and .hi[Mean]: 250.5 rebounds

---
# Clicker Question
	
What is the sample average of the participants?
	
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      <th colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Sample of individuals</span></th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">BMI</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Drinks per week</th>
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    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
<td class="gt_row gt_left">3 drinks</td></tr>
    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">60</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
<td class="gt_row gt_left">1 drink</td></tr>
    <tr><td class="gt_row gt_right">18</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
<td class="gt_row gt_left">6 drinks</td></tr>
    <tr><td class="gt_row gt_right">57</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">31.88</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">56</td>
<td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">male</td>
<td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">42.80</td>
<td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">3 drinks</td></tr>
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---
# Clicker Question
	
Which measure of central tendency best describes the age of participants?

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      <th colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Sample of individuals</span></th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
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    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
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    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
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<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
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<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
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<td class="gt_row gt_left">female</td>
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</div>

<ol type="a">
	<li>Median</li>
	<li>Mean</li>
</ol>

---
# Measuring Variability
	
	
Measures of central tendency do not tell the whole story. To further characterize the distribution, we need to know how the data is spread out
		      
- Quartiles
- Variance

---
# Variability: Quartiles
	
	
- Measure of center alone can be misleading
- How to calculate quartiles:
		      
- Arrange observations in increasing order and locate .hi.red_pink[median] 
- The .hi.green[first quartile] is the median of the observations located to the left of the median
- The .hi.green[third quartile] is the median of observations located to the right of the median
              
<img src="data:image/png;base64,#quartiles.png" width="50%" style="display: block; margin: auto;" />

---
# Boxplots
	
.hi.blue[five-number summary]: smallest observation (minimum), the first quartile, the median, the third quartile, and the largest observation (maximum)

We can use the .hi.purple[boxplot] using this five number summary to display quantitative data

- How to make a boxplot:
		      
- A central box spans the first and third quartiles 
- A line in the box marks the median
- Line extends from the box out to the smallest and largest observations

---
# Boxplots

---
# Interquartile Range
	
	
The .hi.turquoise[interquartile range], IQR, is the distance between the first and third quartiles
		      
- IQR = `$Q_3 - Q_1$`
- The IQR measures the spread of the data and it also helps to identify outliers
	      
Rule for outliers:
		      
- An observation is an outlier if it falls more than `$1.5 \times IQR$` above the third quartile or below the first

---
# Variability: Variance
	
	
.hi.purple[Variance]: denoted, `$s^2$`, measures how "spread out" the data are on average
		     
`$$s^2 = \frac{(x_1-{\color{orange}\bar{x}})^2 + (x_2-{\color{orange}\bar{x}})^2 + .... + (x_n - {\color{orange}\bar{x}})^2}{n-1},$$`
or more compactly

$$
s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - {\color{orange}\bar{x}})^2
$$
.hi.purple[Standard deviation]: looks at how far each observation is from the mean; square root of the variance

`$$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2}$$`

---
# Visualizing Standard Deviation
	
	
<img src="data:image/png;base64,#ch2_files/figure-html/multiple-vars-1.svg" width="90%" style="display: block; margin: auto;" />

---
# Practice Question

Calculate the standard deviation of age?

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  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#fdystepxpp .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#fdystepxpp .gt_left {
  text-align: left;
}

#fdystepxpp .gt_center {
  text-align: center;
}

#fdystepxpp .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#fdystepxpp .gt_font_normal {
  font-weight: normal;
}

#fdystepxpp .gt_font_bold {
  font-weight: bold;
}

#fdystepxpp .gt_font_italic {
  font-style: italic;
}

#fdystepxpp .gt_super {
  font-size: 65%;
}

#fdystepxpp .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Sample of individuals</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Sex</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">BMI</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Drinks per week</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
<td class="gt_row gt_left">3 drinks</td></tr>
    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">60</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
<td class="gt_row gt_left">1 drink</td></tr>
    <tr><td class="gt_row gt_right">18</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
<td class="gt_row gt_left">6 drinks</td></tr>
    <tr><td class="gt_row gt_right">57</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">31.88</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">56</td>
<td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">male</td>
<td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">42.80</td>
<td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">3 drinks</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Properties of Standard Deviation, `$s$`
	
	
- `$n-1$` is referred to as the degrees of freedom
- `$s$` measures variability about the mean
- `$s$` is always greater than or equal to zero, but usually `$> 0$`
  - When would it be `$=0$`?
- As observations become more variable, `$s$` gets larger
- `$s$` is not resistant in the same way the sample mean is not resistant; a few outliers can change it a lot.

---
# Summary of Summary Statistics

Two basic ways to summarize the center and spread of a distribution
	
- Mean and standard deviation (or variance)

- The five-number summary

.hi.slate[When to Use Which]
	
Use `$\bar{x}$` and `$s$` when the distribution is reasonably symmetric and free of outliers
		      
Use five-number summary if distribution is skewed, or has outliers

---
# Greek Letters and Statistics

.pull-left[
.hi.purple[Greek Letters]

- Greek letters like `$\mu$` and `$\sigma^2$` represent the truth about the population.
]
.pull-right[
.hi.green[Latin Letters]

- Latin lettes like `$\bar{x}$` and `$s^2$` are calculations that represent guesses (estimates) at the population values.
]

The goal for the class is for the latin letters to be good guesses for the greek letters:

$$
	{\color{green}\text{Data}} \longrightarrow {\color{green}\text{Calculation}} \longrightarrow {\color{green}\text{Estimates}} \longrightarrow^{hopefully!} {\color{purple}\text{Truth}}
$$

For example, 
$$
	{\color{green}X} \longrightarrow {\color{green} \frac{1}{n} \sum_{i=1}^n X_i} \longrightarrow {\color{green}\bar{x}} \longrightarrow^{hopefullly!} {\color{purple}\mu}
$$

---
# Install R and R Studio
	
.hi[Download R:] [https://www.r-project.org/](https://www.r-project.org/)

- Click "download R" link under "Getting Started"
- Select a CRAN location (mirror site) and click link
- I selected the UC Berkeley one, pick one in USA
- Click on "Download R for Mac/Windows/etc" link at top of page
- Click on package to download, under "Latest Release"
- Save the .pkg file, double click open, and follow instructions

.hi[Download RStudio:] [https://www.rstudio.com/](https://www.rstudio.com/)

- \url{www.rstudio.com} and click "Download RStudio"
- Click on "download RStudio Desktop"

---
# How to use R
	
		
<img src="data:image/png;base64,#r.png" width="90%" style="display: block; margin: auto;" />